Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1340 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,6)(3,7,8,5)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26,31,44,30,35,29,36)(27,33,43,32,38,49,39,47)(28,34,45,55,41,51,42,50)(37,48,54,46,52,56,40,53);(1,3,4,8)(2,5,6,7)(11,17)(12,20)(14,21)(15,19)(25,27,30,38)(26,32,35,47)(28,40,41,54)(29,43,31,39)(33,36,49,44)(34,48,51,56)(37,45,52,42)(46,55,53,50);(2,6)(5,7)(9,11,13,17)(10,14,16,21)(12,18,20,23)(15,22,19,24)(25,27)(26,33)(28,37)(29,39)(30,38)(31,43)(32,44)(34,48)(35,49)(36,47)(40,42)(41,52)(45,54)(46,55)(50,53)(51,56);(3,8)(5,7)(11,17)(14,21)(18,23)(22,24)(25,28,31,45,30,41,29,42)(26,34,44,55,35,51,36,50)(27,37,43,54,38,52,39,40)(32,46,49,56,47,53,33,48);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(26,35)(27,38)(28,41)(36,44)(39,43)(42,45)(46,53)(48,56)Polycyclic presentation:
2^9|a^2=fh,b^2=gi,c^2=hi,d^2=fgi,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7105162);